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If 2^x - 2^x-2 = 3(2^13), what is the value of x

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If 2^x - 2^x-2 = 3(2^13), what is the value of x [#permalink]

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New post 11 Dec 2008, 10:13
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

Here is an exponent problem from GMATPrep, Practice Test 1:

If 2^x - 2^x-2 = 3(2^13), what is the value of x?

a) 9
b) 11
c) 13
d) 15
e) 17

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Re: Exponent problem [#permalink]

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New post 11 Dec 2008, 11:26
D) 15.

I figured this out by using smaller numbers and seeing a pattern.

Take \(2^6 - 2^4\) = 64 - 16 = 48. Divide this by 3 (as we see in the problem) and you get 3 * 16. Well, 16 is the same as \(2^4\).

if 2^6 - 2^4 = 3(2^z). I set the exponent to z because we don't know what it will be. We now only have 1 variable, z, because we gave x the value of 6.

64 - 16 = 3(2^z)
48 = 3(2^z)
48/3 = 2^z
16 = 2^z

2*8 = 2^z
2*2*4 = 2^z
2*2*2*2 = 2^z,

2^4 = 2^z

So apply this back to the original problem

We see that 2^4 was the same as our lowest number in \(2^6 - 2^4\), so 3(2^z) must be equal to \(2^{x-2}\)

So in the original, take \(3(2^{13}) = 3(2^{x-2})\)

\(2^{13} = 2^{x-2}\)...so 13 = x-2. Add 2 to both sides means 15 = x

We have \(3(2^{13})\). If the 13 in here = x-2, then x = 15.

above720 wrote:
Here is an exponent problem from GMATPrep, Practice Test 1:

If 2^x - 2^x-2 = 3(2^13), what is the value of x?

a) 9
b) 11
c) 13
d) 15
e) 17

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**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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Re: Exponent problem [#permalink]

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New post 11 Dec 2008, 13:32
15

solved from the answers

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Re: Exponent problem [#permalink]

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New post 12 Dec 2008, 13:41
I would also go with D. But the solution is different.

2 ^x - 2 ^ (x-2) = 3 ( 2 ^ 13)

==> 2 ^ x - ((2 ^ x) / 2 ^ 2) = 3 (2 ^ 13)
==> 2 ^x(1-1/(2 ^2) = 3 (2 ^ 13)
==> (2 ^ x) (3 / 2^2 ) = 3 (2 ^ 13)
==> 2 ^x = 2 ^ 15

So x = 15.

Kudos [?]: 361 [0], given: 15

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Re: Exponent problem [#permalink]

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New post 13 Dec 2008, 01:02
Same answer but with a different explanation.

Left side = \(2^(x-2)*(2^2-1)
= 2^(x-2)*3\)
Right side = 3*2^12
Hence, x-2 = 13 or, x = 15.

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Re: Exponent problem   [#permalink] 13 Dec 2008, 01:02
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If 2^x - 2^x-2 = 3(2^13), what is the value of x

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